Non-local initial-boundary value problem for a degenerate fourth-order equation with a fractional Gerasimov-Caputo derivative

A. K. Urinov ; D. A. Usmonov

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 42 (2023) no. 1, pp. 123-139

Résumé

Recently, initial-boundary problems in a rectangular domain for differential equations in partial derivatives of both even and odd order have been intensively studied. In this case, non-degenerate equations or equations that degenerate on one side of the quadrilateral are taken as the object of study. But initialboundary problems (both local and non-local) for equations with two or three lines of degeneracy remain unexplored. In this paper, in a rectangular domain, a fourth-order equation degene-rating on three sides of the rectangular and contains the Gerasimov-Caputo fractional diffe-rentiation operator has been considered. For this equation, an initial-boundary problem is formulated and investigated, with non-local conditions connecting the values of the desired function and its derivatives up to the third order (inclusive), taken on the sides of the rectangle. From the beginning, the uniqueness of the solution of the formulated problem was proved by the method of energy integrals. Then, the spectral problem that arises when applying the Fourier method based on the separation of variables to the considered initial-boundary problem has been investigated. The Green's function of the spectral problem was constructed, with the help of which it is equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel, which implies the existence of a countable number of eigenvalues and eigenfunctions of the spectral problem. A theorem is proved for expanding a given function into a uniformly convergent series in terms of a system of eigenfunctions. Using the found integral equation and Mercer's theorem, we prove the uniform convergence of some bilinear series depending on the found eigenfunctions. The order of the Fourier coeffi-cients have been established. The solution of the considered is written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation is studied. An estimate for solution to problem is obtained, from which follows its continuous dependence on the given functions.

Détail

Keywords: degenerate fourth order equation, initial boundary value problem, method of separation of variables, spectral problem, Green's function, integral equation, uniqueness and stability of the solution.
Mots-clés : existence

Bibliographie

[1] Dzhrbashyan M. M., Nersesyan A. B., “Drobnye proizvodnye i zadacha Koshi dlya differentsialnykh uravnenii drobnogo poryadka”, Izv. AN Arm SSR, 3:1 (1968), 3-29 | Zbl

[2] Dzhrbashyan M. M., “Kraevaya zadacha dlya differentsialnogo operatora drobnogo poryadka tipa Shturma - Liuvillya”, Izv. AN ArmSSR. Mat, 5:2 (1970), 71-96 | Zbl

[3] Nakhushev A. M., “Zadacha Shturma-Liuvillya dlya obyknovennogo differentsialnogo uravneniya vtorogo poryadka s drobnymi proizvodnymi v mladshikh chlenakh”, Dokl. AN SSSR, 234:2 (1977), 308-311 | Zbl

[4] Aleroev T. S., “K probleme o nulyakh funktsii Mittaga-Lefflera i spektre odnogo differentsialnogo operatora drobnogo poryadka”, Differents.uravneniya, 36:9 (2000), 1278–1279 | Zbl

[5] Pskhu A. V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005, 199 pp.

[6] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003, 272 pp.

[7] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp.

[8] Berdyshev A. S., Cabada A., Kadirkulov B. J., “The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional differential operator”, Computers and Mathematics with Applications, 62 (2011), 3884-3893 | DOI | Zbl

[9] Berdyshev A. S, Kadirkulov B. J., “A Samarskii-Ionkin problem for two-dimensionalparabolic equation with the Caputo fractional differential operator”, International Journal of Pure and Applied Mathematics, 113:4 (2017), 53-64

[10] Kerbal S., Kadirkulov B. J., Kirane M., “Direct and inverse problems for a Samarskii-Ionkin type problem for a two dimensional fractional parabolic equation”, Progr. Fract. Differ. Appl, 3 (2018), 147-160 | DOI

[11] Aziz S., Malik S. A., “Identifcation of an unknown source term for a time fractional fourth-order parabolic equation”, Electron. J. Differ. Equat., 293 (2016), 1–20

[12] Berdyshev A. S., Kadirkulov B. Zh., “Ob odnoi nelokalnoi zadache dlya parabolicheskogo uravneniya chetvertogo poryadka s drobnym operatorom Dzhrbashyana–Nersesyana”, Differentsialnye uravneniya, 52:1 (2016), 123–127 | DOI | Zbl

[13] Karasheva L. L., “Zadacha v polupolose dlya parabolicheskogo uravneniya chetvertogo poryadka s operatorom Rimana – Liuvillya po vremennoi peremennoi”, Izvestiya Kabardino-Balkarskogo nauchnogo tsentra RAN, 5:91 (2019), 21-29

[14] Yuldashev T. K., Kadirkulov B. J., “Nonlocal problem for a mixed typefourth-order differential equation with Hilfer fractional operator”, Ural mathematical journal, 6:1 (2020), 153–167 | DOI | Zbl

[15] Yuldashev T. K., Kadirkulov B.J., “Inverse boundary value problem for a fractional differential equations of mixed type with integral rede?nition conditions”, Lobachevskii Journal of Mathematics, 42:3 (2021), 649–662 | DOI | Zbl

[16] Ashurov R., Umarov S., “Determination of the order of fractional derivative for subdiffusion equations”, Fract. Calc. Appl. Anal, 23:6 (2020), 1647–1662 | DOI | Zbl

[17] Ashurov R., Fayziev Y, “Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation”, Mathematical Notes, 110:6 (2021), 842–852 | DOI | Zbl

[18] Karimov D. Kh., Kasimova M., “Smeshannaya zadacha dlya lineinogo uravneniya chetvertogo poryadka, vyrozhdayuschegosya na granitse oblasti”, Izv. AN UzSSR, ser. fiz. -mat. nauk, 2 (1968), 27-31

[19] Baikuziev K. B., Kasimova M., “Smeshannaya zadacha dlya uravneniya chetvertogo poryadka, vyrozhdayuschegosya na granitse oblasti”, Izv. AN UzSSR, ser. fiz. -mat. nauk, 5 (1968), 7-12 | Zbl

[20] Kasimova M, “Smeshannaya zadacha dlya lineinogo uravneniya chetverogo poryadka, vyrozhdayuschegosya na granitse oblasti”, Izv. AN UzSSR, ser. fiz. -mat. nauk, 5 (1968), 35-39

[21] Beitmen G., Erdeii A, Vysshie transtsendentnye funktsii, v. 1, Nauka, M., 1965, 296 pp.

[22] Naimark M. A, Lineinye differentsialnye operatory, Nauka, M., 1969, 528 pp.

[23] Mikhlin S. G, Lektsii po lineinym integralnym uravneniyam, Fizmatlit, Moskva, 1959, 232 pp.

[24] Boudabsa L., Simon T, “Some Properties of the Kilbas-Saigo Function”, Mathematics, 9:217 (2021)